Answer

The larger gear rotates through an angle of $156^{\circ}$

Work Step by Step

We can convert $300^{\circ}$ to radians:
$(300^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 5.236~radians$
When the gears rotate, they will rotate through the same arc length. The radius of the smaller gear is 3.7 cm. We can find the arc length $d$ of the smaller gear when it rotates through an angle of $5.236~rad$:
$d = (5.236~rad)(3.7~cm)$
$d = 19.373~cm$
The larger gear will rotate along this same arc length. The radius of the larger gear is 7.1 cm. We can find the angle $x$ through which the larger gear rotates:
$x = \frac{19.373~cm}{7.1~cm}$
$x = 2.73~rad$
We can convert $2.73~rad$ to degrees:
$(2.73~rad)(\frac{180^{\circ}}{\pi~rad}) = 156^{\circ}$
The larger gear rotates through an angle of $156^{\circ}$